Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(f(x)=x^{3}-4 x\)

Step-by-Step Solution

Answer

The function \(f(x) = x^{3} - 4x\) is symmetric with respect to the origin (it's an odd function), and it intersects the \(x\)-axis at three points: \(x = 0\), \(x = -2\), and \(x = 2\).

Step 1: Graphing the Function

Using a graphing utility, plot the function \(f(x) = x^3 - 4x\). This will give us a visual representation of the function and help us identify any symmetry and \(x\)-intercepts.

Step 2: Identifying Symmetry

Look at the graph and check if it's symmetric with respect to the \(y\)-axis (even function) or the origin (odd function). If the left half of the graph is a mirror image of the right half, then the function is even and it's symmetric with respect to the \(y\)-axis. If the graph is rotated 180 degrees about the origin and remains unchanged, then the function is odd. In this case, the graph of the function \(f(x) = x^3 - 4x\) is not symmetric about the \(y\)-axis, but it is symmetric about the origin hence it is an odd function.

Step 3: Finding the Intercepts

The \(x\)-intercepts are the points where the graph intersects the \(x\)-axis. To solve for the intercepts algebraically, we set \(f(x) = 0\), so \(x^3 - 4x = 0\). From this equation, we get three \(x\)-intercepts, \(x = 0, x = -2, x = 2 \)

Key Concepts

Symmetry in Graphs

Symmetry in graphs helps us understand how a graph is mirrored across an axis or rotated around a point. When studying functions, identifying symmetry can significantly simplify their analysis. Specifically, if a function is symmetrical about the y-axis, each point \( (x, y) \) on the function has a mirrored point \( (-x, y) \) also on the function. Conversely, symmetry about the x-axis means each point \( (x, y) \) is paired with \( (x, -y) \) on the graph.

X-Intercepts

X-intercepts are where a graph crosses or touches the x-axis. These points are crucial in understanding the behavior of a function and can indicate the presence of roots, or solutions, to the equation \( f(x) = 0 \). You can find the x-intercepts by solving this equation. For example, with the function \( f(x) = x^3 - 4x \), setting \( f(x) = 0 \) yields the intercepts as \( x = 0, -2, \text{and} 2 \) which correspond to the points where the function crosses the x-axis. These intercepts divide the graph into sections and can reveal intervals where the function is positive or negative.

Odd Functions

A function is classified as an odd function if its graph exhibits rotational symmetry around the origin. This symmetry is described mathematically by the property \( f(-x) = -f(x) \). For any x-value, the function's value is negated when the sign of x is reversed. The classical visual representation of an odd function is when rotating the graph by 180 degrees results in the same graph. For instance, the given function \( f(x) = x^3 - 4x \) is odd because it fulfills this criterion. Identifying a function as an odd function immediately provides a plethora of information: there will be a symmetry about the origin, and frequently, the function will pass through the origin itself.